Bifurcational Aspects of Parametric Resonance

Golubitsky

Vegter³

2003

Nonlinearity

Resonance tongues and their boundaries are studied for nondegenerate and (certain) degenerate Hopf bifurcations of maps using singularity theory methods of equivariant contact equivalence and universal unfoldings. We recover the standard theory of tongues (the nondegenerate case) in a straightforward way and we find certain surprises in the tongue boundary structure when degeneracies are present. For example, the tongue boundaries at degenerate singularities in weak resonance are much blunter than expected from the nondegenerate theory. Also at a semi-global level we find 'pockets' or 'flames' that can be understood in terms of the swallowtail catastrophe.

Section: Background and Sketch Of Resultssupporting

confidence: 61%

The geometry of resonance tongues: a singularity theory approach

Golubitsky

Vegter³

2003

Nonlinearity

“…For similar approaches e.g. see Arnold [6], Broer [10,11], Broer and Vegter [16], Duistermaat [19] or Van der Meer [46].…”

Section: H 8h =Tf' Y-mentioning

confidence: 99%

“…Therefore, instead, we present a Formal Normal Form permitting the formulation of a perturbation problem related to the 2-dimensional situation of Section 2. The relevant normal form theory started with Poincar6 [35] and Birkhoff [7], for more recent references see Gustavson [24], Moser [30,31], Takens [43,44], Arnold [5], Sanders [39], Broer [11,10], Sanders and Verhulst [40], Van der Meer [46] and Broer and Vegter [16].…”

Section: A Hamiltonian Normal Form: Formal Reduction To 1 Degree Of Fmentioning

confidence: 99%

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A normally elliptic Hamiltonian bifurcation

Chow

Kim

et al. 1993

Z. angew. Math. Phys.

Self Cite

The unfolding theory, to be developed below, thenThe conjugate pair of imaginary eigenvalues gives rise to a formal rotational symmetry in all unfoldings, i.e. a rotational symmetry in their Taylor series. Here the series is considered in dependence on both the phase space variables and the parameters. This is an application of Normal Form Theory, where the terms of the formal power series are changed by canonical coordinate transformations in an inductive process. The symmetry, thus obtained, enables a formal reduction to 1 degree of freedom, around an equilibrium point with a double eigenvalue 0. For similar approaches see some of the above references, also compare e.g. Arnold [6], Takens [43,44], Broer [10,11], Golubitsky and Stewart [23] and Broer and Vegter [16]. This Normal Form Theory will be presented in Section 3.In Section 2, we begin studying the 1 degree of freedom 'backbone'-problem in its own right. Since in the plane the integral curves of the systems are the level sets of the Hamiltonian functions, here the problem reduces to Singularity Theory, e.g. In Section 4 the connection is made between the planar reduction of the symmetric system and the 'backbone'-system of Section 2. It turns out that the formal integral, obtained by normalization, is a distinguished parameter in the sense of Golubitsky and Schaeffer [51] and Schecter [41], also compare Wassermann [52]. Now Singularity Theory yields new normal forms, that are polynomial of degree 3, at least in the phase-space variables. We note that here the normalizing transformations no longer need to be canonical. Technical details from Singularity Theory have been collected in an appendix (Section 7). After this we suspend, or dereduce, to the original 4-dimensional setting, so obtaining an integrable, i.e. rotationally symmetric, approximation of the original unfolding.Thus we obtain a perturbation problem, similar to e.g. Broer [10,12] or Braaksma and Broer [8]. The perturbation term is of arbitrarily high order, both in the phase space variables and the parameters. This problem is briefly addressed in Section 5.Remarks. (i) Although in the original family only generic restrictions are imposed on the lower order terms, by Normal Form Theory and Singularity Theory, the corresponding unfolding is reduced to an arbitrarily flat perturbation of a normal form, completely determined by a 1 degree of freedom system, which is polynomial of degree 3. This polynomial character Vol. 44, 1993 A normally elliptic Hamiltonian bifurcation 391 may explain why certain 'integrable' characteristics in the unfoldings are so persistent. For a related comment we refer to Verhulst [49].(ii) Our approach differs from e.g. Van der Meer [46], also compare Duistermaat [19]. In [46,19] the dynamics is reduced by the energy-momentum map as well as the Moser-Weinstein method. This gives information related to specific periodic solutions, that is also important for the dynamics as a whole. Instead, we just factor out a formal rotational symmetry, and so seem to get a more ...

The Inverted Pendulum: A Singularity Theory Approach

Journal of Differential Equations

Hoveijn

Noort

et al. 1999

Self Cite

The inverted pendulum with small parametric forcing is considered as an example of a wider class of parametrically forced Hamiltonian systems. The qualitative dynamics of the Poincare map corresponding to the central periodic solution is studied via an approximating integrable normal form. At bifurcation points we construct local universal models in the appropriate symmetry context, using equivariant singularity theory. In this context, structural stability can be proved under generic conditions. Academic Press